Equivalent Sets have Equal Cardinal Numbers

Theorem

Let $S$ and $T$ be sets.

Let $\card S$ denote the cardinal number of $S$.

Then:

$S \sim T \implies \card S = \card T$

Proof

Let $x$ be an arbitrary set that is an ordinal:

\(\ds S\)

\(\sim\)

\(\ds T\)

by hypothesis

\(\ds \leadsto \ \ \)

\(\ds S \sim x\)

\(\iff\)

\(\ds T \sim x\)

Set Equivalence behaves like Equivalence Relation

\(\ds \leadsto \ \ \)

\(\ds \set {x \in \On : S \sim x}\)

\(=\)

\(\ds \set {x \in \On : T \sim x}\)

Definition of Class Equality

\(\ds \leadsto \ \ \)

\(\ds \bigcap \set {x \in \On : S \sim x}\)

\(=\)

\(\ds \bigcap \set {x \in \On : T \sim x}\)

Substitutivity of Class Equality

\(\ds \leadsto \ \ \)

\(\ds \card S\)

\(=\)

\(\ds \card t\)

Definition of Cardinal Number

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.14$